Solve (a+1)(a+8)-(a-6)(a-6)

$$( a + 1 ) ( a + 8 ) - ( a - 6 ) ( a - 6 )$$

Evaluate

$$21a-28$$

Solution Steps

Multiply a-6 and a-6 to get \left(a-6\right)^{2}.

$$\left(a+1\right)\left(a+8\right)-\left(a-6\right)^{2}$$

Apply the distributive property by multiplying each term of a+1 by each term of a+8.

$$a^{2}+8a+a+8-\left(a-6\right)^{2}$$

Combine 8a and a to get 9a.

$$a^{2}+9a+8-\left(a-6\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-6\right)^{2}.

$$a^{2}+9a+8-\left(a^{2}-12a+36\right)$$

To find the opposite of a^{2}-12a+36, find the opposite of each term.

$$a^{2}+9a+8-a^{2}-\left(-12a\right)-36$$

The opposite of -12a is 12a.

$$a^{2}+9a+8-a^{2}+12a-36$$

Combine a^{2} and -a^{2} to get 0.

$$9a+8+12a-36$$

Combine 9a and 12a to get 21a.

$$21a+8-36$$

Subtract 36 from 8 to get -28.

$$21a-28$$

Expand

$$21a-28$$

Solution Steps

Multiply a-6 and a-6 to get \left(a-6\right)^{2}.

$$\left(a+1\right)\left(a+8\right)-\left(a-6\right)^{2}$$

Apply the distributive property by multiplying each term of a+1 by each term of a+8.

$$a^{2}+8a+a+8-\left(a-6\right)^{2}$$

Combine 8a and a to get 9a.

$$a^{2}+9a+8-\left(a-6\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-6\right)^{2}.

$$a^{2}+9a+8-\left(a^{2}-12a+36\right)$$

To find the opposite of a^{2}-12a+36, find the opposite of each term.

$$a^{2}+9a+8-a^{2}-\left(-12a\right)-36$$

The opposite of -12a is 12a.

$$a^{2}+9a+8-a^{2}+12a-36$$

Combine a^{2} and -a^{2} to get 0.

$$9a+8+12a-36$$

Combine 9a and 12a to get 21a.

$$21a+8-36$$

Subtract 36 from 8 to get -28.

$$21a-28$$